Joint Variation

Joint Variation involves at least three variables. For example, in gas laws, the pressure P is directly proportional to temperature T and inversely proportional to volume V. 

i.e. \(\scriptsize P \propto \normalsize \frac{T}{V} \)

then \(\scriptsize P = \normalsize \frac{KT}{V} \)

where k is a constant. 

This means P varies jointly as the quotient of T and V. 

Example 9.1.1:

If \(\scriptsize a \propto bc \) and a = 5, when b = 4, c = 2 

(a) write down the equation connecting these variables. 
(b) find a when b = 16, c = 1 
(c) find c when a = 10, b = 20. 

Solution

(a)

\(\scriptsize a \propto bc \)

a = kbc  where k = constant 

k = \( \frac{a}{bc} \)

 a = 5, when b = 4, c = 2 

Therefore, k = \( \frac{5}{4 \: \times \: 2} \)

k = \( \frac{5}{8} \)

a = kbc

∴ the equation connecting the variables is: a = \( \frac{5}{8} \scriptsize bc \)

(b)

find a when b = 16,  c = 1